I want to show the uniform convergence of the series
$$\sum_{n=1}^{\infty} \frac{x}{n(1+nx^2)}$$

It's easy to show the uniform convergence for $|x| > \epsilon$, for any positive $\epsilon$, by using Weierstrass M-test. But I'm having trouble showing the uniform convergence in a neighborhood of zero. Any hints?

1 Answer
1

Let $f_n(x):=\frac x{n(1+nx^2)}$. Then $$f'_n(x)=\frac 1n\frac{1+nx^2-2nxx}{(1+nx^2)^2}=\frac 1n\frac{1-nx^2}{(1+nx^2)^2}$$
so $f_n$ reaches its maximum on $\mathbb R_+$ at $x_n:=\frac 1{\sqrt n}$. We have
$f_n(x_n)=\frac{n^{-1/2}}{n(1+n(n^{-1/2})^2)}=\frac 1{2n^{3/2}}$, and since $\sum_{n\geq 1}\frac 1{n^{3/2}}$ is convergent we conclude that the series $\sum_n f_n$ is normally convergent on $\mathbb R_+$, and since each $f_n$ is odd we have the normal hence uniform convergence on $\mathbb R$.